Subgroups — #10695210

Order 768.

506 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#1744	C₂	1	1	yes	maximal, characteristic
3	#69394	C₂	1	1	yes	maximal, characteristic
4	#67655	C₂	1	4	yes	maximal
5	#9874	—	3	1	no	maximal, Sylow
6	C₂×#869	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆	—	2	1	no	—
8	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4935	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₃	—	2	1	no	—
12	C₄×C₄⋊Dic₃	C₂²	1	2	yes	—
13	#4929	—	2	1	no	—
14	#4928	—	2	1	no	—
15	#4928	C₄	1	2	yes	—
16	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
17	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×#336	—	2	1	no	—
20	C₃×#336	—	2	1	no	—
21	C₄×C₄⋊Q₈	—	3	1	no	—
22	#1842	—	3	1	no	—
23	#1744	D₃	1	1	yes	characteristic
24	#1279	—	3	4	no	—
25	C₂²×Dic₆	—	2	1	no	—
26	C₂²×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₃	—	2	1	no	—
35	C₂×C₄⋊Dic₃	—	2	1	no	—
36	C₂×C₄⋊Dic₃	—	2	1	no	—
37	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
39	C₂×Dic₃⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₃	—	2	2	no	—
41	C₂²×C₃⋊C₈	—	2	2	no	—
42	#869	—	2	4	no	—
43	C₄×Dic₆	—	4	2	no	—
44	C₄×Dic₆	—	4	2	no	—
45	C₃×C₄×C₈	—	2	2	no	—
46	C₃×C₄×C₈	Q₈	1	4	yes	complemented
47	C₃⋊#95	—	2	2	no	—
48	C₃⋊#95	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	—	2	4	no	—
50	C₄⋊(C₃⋊C₈)	—	4	2	no	—
51	C₄⋊(C₃⋊C₈)	—	4	2	no	—
52	C₂×C₄⋊Q₈	—	3	1	no	—
53	C₂×C₄×Q₈	—	6	1	no	—
54	C₂×C₄⋊C₈	—	3	2	no	—
55	C₂×C₄⋊C₈	—	6	1	no	—
56	C₂×C₄⋊C₈	—	6	1	no	—
57	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
58	#391	—	6	1	no	—
59	#389	—	3	2	no	—
60	#389	—	6	1	no	—
61	#384	—	3	1	no	—
62	C₄×C₄⋊C₄	—	3	2	no	—
63	C₄×C₄⋊C₄	—	6	1	no	—
64	C₄³	C₂×D₃	1	1	yes	characteristic
65	#336	—	2	1	no	—
66	#336	—	2	1	no	—
67	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₃	—	2	2	no	—
75	C₂²×Dic₃	—	2	2	no	—
76	C₂×Dic₆	—	2	4	no	—
77	C₂×Dic₆	—	4	2	no	—
78	C₂×Dic₆	—	4	2	no	—
79	C₂×C₃×C₈	—	2	4	no	—
80	C₂×C₃×C₈	—	4	2	no	—
81	C₂×C₃×C₈	—	4	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	4	1	no	—
85	C₃×C₄²	—	4	1	no	—
86	C₃×C₄²	D₈	1	2	yes	—
87	C₃×C₄²	SD₁₆	1	2	yes	—
88	C₃×C₄²	SD₁₆	1	2	yes	—
89	C₃×C₄²	Q₁₆	1	2	yes	—
90	C₃×C₄²	C₂×Q₈	1	2	yes	—
91	C₃×C₄²	#41	1	2	yes	—
92	C₄⋊Dic₃	—	2	4	no	—
93	C₄⋊Dic₃	—	2	2	no	—
94	C₄⋊Dic₃	—	2	4	no	—
95	C₄⋊Dic₃	—	4	2	no	—
96	C₄⋊Dic₃	—	4	2	no	—
97	C₄⋊Dic₃	M₁₆	1	4	yes	—
98	Dic₃⋊C₄	—	4	4	no	—
99	C₄×Dic₃	—	4	2	no	—
100	C₄×Dic₃	—	4	2	no	—
101	C₂×C₃⋊C₈	—	4	2	no	—
102	C₂×C₃⋊C₈	—	4	2	no	—
103	C₂×C₃⋊C₈	—	4	2	no	—
104	C₂²×Q₈	—	6	1	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	—	6	2	no	—
107	C₄⋊Q₈	—	6	4	no	—
108	C₄×Q₈	—	12	2	no	—
109	C₄×Q₈	—	12	2	no	—
110	C₂×C₄⋊C₄	—	3	2	no	—
111	C₂×C₄⋊C₄	—	3	2	no	—
112	C₂×C₄⋊C₄	—	6	1	no	—
113	C₂×C₄⋊C₄	—	6	1	no	—
114	C₂×C₄⋊C₄	—	6	2	no	—
115	C₂×C₄⋊C₄	—	6	1	no	—
116	C₂×C₄²	—	6	2	no	—
117	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
118	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
119	C₂×C₄²	D₁₂	1	1	yes	characteristic
120	C₂×C₄²	D₁₂	1	1	yes	characteristic
121	C₂×C₄²	#67	1	1	yes	characteristic
122	C₂×C₄²	#67	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
124	C₄⋊C₈	—	6	4	no	—
125	C₄⋊C₈	—	12	2	no	—
126	C₄⋊C₈	—	12	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	Dic₆	1	4	yes	complemented
129	#95	—	6	2	no	—
130	#95	—	6	2	no	—
131	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	#102	1	2	yes	—
143	C₂×C₃×C₄	#102	1	2	yes	—
144	C₂×C₃×C₄	#103	1	2	yes	—
145	C₂×C₃×C₄	#103	1	2	yes	—
146	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
147	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₃×C₄	#124	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
151	C₂×C₃×C₄	#131	1	2	yes	—
152	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
156	C₂×Dic₃	—	2	4	no	—
157	C₂×Dic₃	—	4	2	no	—
158	C₂×Dic₃	—	4	2	no	—
159	C₂×Dic₃	—	4	2	no	—
160	C₂×Dic₃	—	4	2	no	—
161	C₂×Dic₃	—	4	2	no	—
162	Dic₆	—	4	8	no	—
163	C₃×C₈	—	4	4	no	—
164	C₃⋊C₈	—	8	2	no	—
165	C₂×Q₈	—	6	4	no	—
166	C₂×Q₈	—	12	2	no	—
167	C₂×Q₈	—	12	2	no	—
168	C₂²×C₄	—	6	2	no	—
169	C₂²×C₄	—	6	2	no	—
170	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
176	C₂²×C₄	C₂×#67	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	12	2	no	—
181	C₂×C₈	—	12	2	no	—
182	C₂×C₈	—	12	2	no	—
183	C₄⋊C₄	—	3	4	no	—
184	C₄⋊C₄	—	6	4	no	—
185	C₄⋊C₄	—	6	2	no	—
186	C₄⋊C₄	—	6	4	no	—
187	C₄⋊C₄	—	12	2	no	—
188	C₄⋊C₄	—	12	4	no	—
189	C₄⋊C₄	—	12	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	—	12	2	no	—
195	C₄²	—	12	2	no	—
196	C₄²	#204	1	2	yes	—
197	C₄²	#204	1	2	yes	—
198	C₄²	D₂₄	1	2	yes	—
199	C₄²	Dic₁₂	1	2	yes	—
200	C₄²	C₂×Dic₆	1	2	yes	—
201	C₄²	#235	1	2	yes	—
202	C₂²×C₃	#391	1	1	yes	characteristic
203	C₂²×C₃	#407	1	1	yes	characteristic
204	C₂²×C₃	#408	1	1	yes	characteristic
205	C₂²×C₃	C₂×#102	1	1	yes	characteristic
206	C₂²×C₃	C₂×#103	1	1	yes	characteristic
207	C₂²×C₃	#421	1	1	yes	characteristic
208	C₂²×C₃	#421	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	#446	1	4	yes	—
215	C₃×C₄	#486	1	1	yes	characteristic
216	C₃×C₄	#487	1	1	yes	characteristic
217	C₃×C₄	#498	1	1	yes	characteristic
218	C₃×C₄	#500	1	1	yes	characteristic
219	Dic₃	—	4	4	no	—
220	Dic₃	—	8	2	no	—
221	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
222	Q₈	—	12	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	6	4	no	—
234	C₂×C₄	—	12	2	no	—
235	C₂×C₄	—	12	2	no	—
236	C₂×C₄	—	12	2	no	—
237	C₂×C₄	—	12	2	no	—
238	C₂×C₄	—	12	2	no	—
239	C₂×C₄	#816	1	2	yes	—
240	C₂×C₄	#816	1	2	yes	—
241	C₂×C₄	#821	1	2	yes	—
242	C₂×C₄	#821	1	2	yes	—
243	C₂×C₄	C₄×Dic₆	1	2	yes	—
244	C₂×C₄	#869	1	1	yes	characteristic
245	C₂×C₄	#875	1	1	yes	characteristic
246	C₂×C₄	C₂×#204	1	1	yes	characteristic
247	C₂×C₄	C₂×#204	1	1	yes	characteristic
248	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
250	C₂×C₄	D₃×M₁₆	1	2	yes	—
251	C₂×C₄	#907	1	2	yes	—
252	C₂×C₄	#924	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	24	2	no	—
255	C₂×C₃	#1480	1	1	yes	characteristic
256	C₂×C₃	#1481	1	1	yes	characteristic
257	C₂×C₃	#1486	1	1	yes	characteristic
258	C₂×C₃	#1487	1	1	yes	characteristic
259	C₂×C₃	#1985	1	1	yes	characteristic
260	C₂×C₃	#1990	1	1	yes	characteristic
261	C₂×C₃	#1995	1	1	yes	characteristic
262	C₂²	#4929	1	1	yes	characteristic
263	C₂²	#5005	1	1	yes	characteristic
264	C₂²	#5005	1	1	yes	characteristic
265	C₂²	C₂×#816	1	1	yes	characteristic
266	C₂²	C₂×#821	1	1	yes	characteristic
267	C₂²	#5126	1	1	yes	characteristic
268	C₂²	#5129	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	12	4	no	—
275	C₄	—	24	2	no	—
276	C₄	#4680	1	1	yes	characteristic
277	C₄	#4681	1	1	yes	characteristic
278	C₄	#4682	1	1	yes	characteristic
279	C₄	#4694	1	1	yes	characteristic
280	C₄	#4701	1	4	yes	—
281	C₃	#9874	1	1	yes	characteristic, complemented, Sylow
282	C₂	#69441	1	1	yes	characteristic
283	C₂	#69777	1	1	yes	characteristic
284	C₂	#69838	1	1	yes	characteristic
285	C₂	#70758	1	1	yes	characteristic
286	C₂	#70762	1	1	yes	characteristic
287	C₂	#70768	1	1	yes	characteristic
288	C₂	#70770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1744	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1744	D₃
hypocenter	C₃	#9874
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
46. C₃×C₄×C₈	Q₈	semidirect
128. C₄×C₈	Dic₆	semidirect
281. C₃	#9874	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#869	2
C₃×#1744	2
#69394	2
#67655	2
#9874	3